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ADMN 3116: Financial Management 1
Lecture 7: Portfolio selection
Anton Miglo
Fall 2014
ADMN 3116, Anton Miglo
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Topics Covered
 Efficient Set of Portfolios
 Sharpe ratio and optimal portfolio
 Optimal portfolio with risk-free asset available
 Excel: Solver
 Additional readings: ch. 10-11 B
ADMN 3116, Anton Miglo
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ADMN 3116, Anton Miglo
Diversification
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Investment mistakes
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1. “Put all eggs in one basket”
2. Superfluous or Naive Diversification
(Diversification for diversification’s sake)
a. Results in difficulty in managing such a
large portfolio
b. Increased costs (Search and transaction)
3. Many investors think that diversification is
always associated with lower risk but also with
lower return
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Correlation
Ontario
Quebec
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Portfolio of two positively correlated assets
Asset B
Asset A
Asset C=1/2A+1/2B
30
30
30
15
15
15
0
0
0
-15
-15
-15
ADMN 3116, Anton Miglo
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Portfolio of two negatively correlated assets
Asset B
Asset A
Asset C=1/2A+1/2B
40
40
40
15
15
15
0
0
0
-10
-10
-10
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Recall: portfolios
 For a portfolio of two assets, A and B, the variance of the return on the
portfolio is:
σ p2  x 2A σ 2A  xB2 σB2  2x A xBCOV(A,B)
σ p2  x 2A σ 2A  xB2 σB2  2x A xBσ A σ BCORR(R ARB )
Where: xA = portfolio weight of asset A
xB = portfolio weight of asset B
such that xA + xB = 1.
(Important: Recall Correlation Definition!)
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The Markowitz Efficient Frontier
 The Markowitz Efficient frontier is the set of portfolios with
the maximum return for a given risk AND the minimum risk
given a return.
 For the plot, the upper left-hand boundary is the Markowitz
efficient frontier.
 All the other possible combinations are inefficient. That is,
investors would not hold these portfolios because they could
get either
 more return for a given level of risk
or
 less risk for a given level of return.
ADMN 3116, Anton Miglo
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Efficient Portfolios with Multiple Assets
E[r]
Efficient
Frontier
Investors
prefer
Asset
Portfolios
Asset 1
of other
Portfolios of
assets
2 Asset 1 and Asset 2
Minimum-Variance
Portfolio
0
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Excel
 Solver
ADMN 3116, Anton Miglo
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Sharpe-Optimal Portfolios
ADMN 3116, Anton Miglo
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Example: Solving for a
Sharpe-Optimal Portfolio
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 From a previous chapter, we know that for a 2-asset portfolio:
Portfolio Return : E(R p )  x sE(R s )  x BE(R B )
Portfolio Variance : σ P2  x S2 σ S2  x B σ B2  2x S x B σ S σ B CORR(R S ,R B )
2
Sharpe Ratio 
E(R p ) - rf
σP

x SE(R S )  x BE(R B ) - rf
x S2 σ S2  x B2 σ B2  2x S x B σ S σ B CORR(R S ,R B )
So, now our job is to choose the weight in asset S that maximizes the
Sharpe Ratio.
We could use calculus to do this, or we could use Excel.
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Example: Using Excel to Solve for
the Sharpe-Optimal Portfolio
Suppose we enter the data (highlighted in yellow) into a
spreadsheet.
We “guess” that Xs = 0.25 is a “good” portfolio.
Using formulas for portfolio return and standard deviation, we
compute Expected Return, Standard Deviation, and a Sharpe
Data
Ratio:
Inputs:
ER(S):
STD(S):
ER(B):
STD(B):
CORR(S,B):
R_f:
ADMN 3116, Anton Miglo
0.12
0.15
0.06
0.10
0.10
0.04
X_S: 0.250
ER(P): 0.075
STD(P): 0.087
Sharpe
Ratio: 0.402
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Example: Using Excel to Solve for
the Sharpe-Optimal Portfolio, Cont.
 Now, we let Excel solve for the weight in portfolio S that maximizes the
Sharpe Ratio.
 We use the Solver, found under Tools.
Solving for the Optimal Sharpe Ratio
Given the data inputs below, we can use the
SOLVER function to find the Maximum Sharpe Ratio:
Data
Inputs:
ER(S):
STD(S):
ER(B):
STD(B):
CORR(A,B):
R_f:
ADMN 3116, Anton Miglo
0.12
0.15
0.06
0.10
0.10
0.04
Changer
Cell:
X_S:
0.700
ER(P):
STD(P):
Sharpe
Ratio:
0.102
0.112
0.553
Target Cell
Well, the “guess” of
0.25 was a tad
low….
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